Godel's disjunctive conclusion ("Some basic theorems on the foundations of
mathematics and their implications" [1951] in Collected Works, vol. III, pp.
304-323):
Either mathematics is incompletable in this sense, that its evident axioms
can never be comprised in a finite rule, that is to say, the human mind
(even within the realm of pure mathematics) infinitely surpasses the powers
of any finite machine, or else there exist absolutely unsolvable Diophantine
problems of the type specified... (p. 310)
Notice that Godels disjunctive conclusion is not the same as the conclusion
drawn by Nagel and Newman (Godel's Proof [1959]), J.R. Lucas ("Mind,
Machines, and Godel" [1961]; "Minds, Machines, and Godel: A Retrospect"
[1996]), or Roger Penrose (The Emperor's New Mind [1989]; Shadows of the
Mind [1994]). All essentially contend that the theorems directly imply that
a Turing machine cannot serve as a model for the human mind. In effect,
they appear to argue thus: Let T be a Turing machine that represents my
own mathematical abilities. Then applying Gödels technique, it is possible
for me to find a proposition that I can prove, but T cannot prove. This
contradicts the statement that T represents me. Therefore, the mind is
not equivalent to a Turing machine.
Putnam ("Minds and Machines" [1960], reprinted in Mind, Language and
Reality: Philosophical Papers [1975], vol. 2, pp. 362-285) has given a
straightforward response to this argument: To find some proposition U from
an arbitrary machine T such that, if T is consistent, T cannot prove U is
not in fact to prove U. Godels more modest conclusion does not suffer from
this flaw.
Zack Purvis